Calculus Examples

$\arctan (x e^{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = x e^{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x e^{x}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [x e^{x}]$

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [x e^{x}]$

Step 1.3

Replace all occurrences of $u$ with $x e^{x}$ .

$\frac{1}{1 + {(x e^{x})}^{2}} \frac{d}{d x} [x e^{x}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x}$ .

$\frac{1}{1 + {(x e^{x})}^{2}} (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x])$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{1}{1 + {(x e^{x})}^{2}} (x e^{x} + e^{x} \frac{d}{d x} [x])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{1 + {(x e^{x})}^{2}} (x e^{x} + e^{x} \cdot 1)$

Step 4.2

Multiply $e^{x}$ by $1$ .

$\frac{1}{1 + {(x e^{x})}^{2}} (x e^{x} + e^{x})$

Step 5

Simplify.

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Step 5.1

Apply the product rule to $x e^{x}$ .

$\frac{1}{1 + x^{2} {(e^{x})}^{2}} (x e^{x} + e^{x})$

Step 5.2

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{1 + x^{2} e^{x \cdot 2}} (x e^{x} + e^{x})$

Step 5.2.2

Move $2$ to the left of $x$ .

$\frac{1}{1 + x^{2} e^{2 x}} (x e^{x} + e^{x})$

Step 5.3

Reorder the factors of $\frac{1}{1 + x^{2} e^{2 x}} (x e^{x} + e^{x})$ .

$(x e^{x} + e^{x}) \frac{1}{1 + x^{2} e^{2 x}}$

Step 5.4

Multiply $x e^{x} + e^{x}$ by $\frac{1}{1 + x^{2} e^{2 x}}$ .

$\frac{x e^{x} + e^{x}}{1 + x^{2} e^{2 x}}$

Step 5.5

Factor $e^{x}$ out of $x e^{x} + e^{x}$ .

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Step 5.5.1

Factor $e^{x}$ out of $x e^{x}$ .

$\frac{e^{x} x + e^{x}}{1 + x^{2} e^{2 x}}$

Step 5.5.2

Multiply by $1$ .

$\frac{e^{x} x + e^{x} \cdot 1}{1 + x^{2} e^{2 x}}$

Step 5.5.3

Factor $e^{x}$ out of $e^{x} x + e^{x} \cdot 1$ .

$\frac{e^{x} (x + 1)}{1 + x^{2} e^{2 x}}$